SubNyquist rate ADC samplingbased compressive ... - Semantic Scholar
WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. (2013) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.2372
RESEARCH ARTICLE
Sub-Nyquist rate ADC sampling-based compressive channel estimation Guan Gui1 , Wei Peng1,2* and Fumiyuki Adachi1 1 2
Tohoku University, Department of Communication Engineering Huazhong University of Science and Technology, Department of Electronics and Information Engineering
ABSTRACT To realize high-speed communication, broadband transmission has become an indispensable technique in the nextgeneration wireless communication systems. Broadband channel is often characterized by the sparse multipath channel model, and significant taps are widely separated in time, and thereby, a large delay spread exists. Accurate channel state information is required for coherent detection. Traditionally, accurate channel estimation can be achieved by sampling the received signal with large delay spread by analog-to-digital converter (ADC) at Nyquist rate and then estimate all of channel taps. However, as the transmission bandwidth increases, the demands of the Nyquist sampling rate already exceed the capabilities of current ADC. In addition, the high-speed ADC is very expensive for ordinary wireless communication. In this paper, we present a novel receiver, which utilizes a sub-Nyquist ADC that samples at much lower rate than the Nyquist one. On the basis of the sampling scheme, we propose a compressive channel estimation method using Dantzig selector algorithm. By comparing with the traditional least square channel estimation, our proposed method not only achieves robust channel estimation but also reduces the cost because low-speed ADC is much cheaper than high-speed one. Computer simulations confirm the effectiveness of our proposed method. Copyright © 2013 John Wiley & Sons, Ltd. KEYWORDS compressive channel estimation; analog-to-digital converter (ADC); sub-Nyquist rate sampling; compressive sensing *Correspondence Wei Peng, Huazhong University of Science and Technology, Department of Electronics and Information Engineering,Luoyu Road 1037,Hongshan District, Wuhan, 430074,China. E-mail: [email protected]
1. INTRODUCTION 1.1. Background and motivation With the increasing number of wireless subscribers, various portable wireless devices, for example, smartphones and laptops, have generated increasing massive data traffic. The demand for high-speed data services is getting more difficult to be satisfied. Broadband transmission is an indispensable technique in the next-generation communication systems [1] and sensor networks [2–5]. However, the broadband signal is susceptible to interference caused by frequency-selective fading. Consequently, accurate channel estimation is required at the receiver for coherent detection. Because the broadband channel is often described by the sparse channel model, sparse channel estimation methods have been proposed to take the advantage of channel sparsity [6–13]. By exploiting the sparse prior
Copyright © 2013 John Wiley & Sons, Ltd.
information, channel estimation performance or spectral efficiency can be improved. Nyquist rate analog-to-digital converter (ADC) sampling system based on traditional sparse channel estimation is shown in Figure 1. As the data rate increases, such a system poses two challenges for the next-generation broadband wireless communication systems. The first challenge is that the requirement of Nyquist sampling speed exceeds the capability of current ADC. Besides, high-speed ADC device becomes very expensive when the sampling rate increases. The second challenge is that Nyquist samplingbased system will reduce the spectrum efficiency because of the large number of training sequence used for channel estimation. Therefore, it is necessary to develop a novel technique to relax the requirement on high-speed ADC sampling in the broadband communication systems. Shannon sampling theorem [14] is one of the fundamental theorems of modern signal processing. Given a
Compressive channel estimation with low-speed ADC
G. Gui, W. Peng and F. Adachi
Figure 1. Communication system using Nyquist sampling rate ADC.
Figure 3. Broadband communication system using sub-Nyquist sampling rate ADC.
continuous signal r.t /, t 2 Œ0; T / whose highest frequency is less than W =2 hertz, Shannon theorem suggests that the signal be sampled uniformly at a rate of W hertz. The signal can be recovered from the samples, given by
1.2. Main contribution
r.t / D
X n2
r
n sinc.W t n/; t 2 Œ0; T / W
(1)
Number of channel sampling taps
However, this well-known approach becomes impractical when the transmission rate or signal bandwidth is large because it is challenging to build a sampling device that operates at a sufficiently high speed. The demands of many modern applications already exceed the capabilities of current technology [15]. Even though recent developments in ADC have increased the sampling rate, state-of-the-art architectures are not yet adequate for the emerging applications. According to the Shannon sampling theorem [14], the number of taps in channel sampling has a linear relation with the transmission bandwidth. And the number of channel sampling taps easily reaches hundreds when its transmission bandwidth becomes large. A simple example of the number of channel sampling taps is depicted in Figure 2.
2
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10
1 2 3
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=1 s =0.5 s =0.1 s
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Transmission bandwidth (MHz) Figure 2. Relation between the number of channel sampling taps with Nyquist rate sampling and transmission bandwidth at different time delay spreads: 1 D 1 s, 2 D 5 s, and 3 D 0:1 s.
In this paper, we investigate compressive channel estimation (CCE) problem on the basis of low-speed ADC working at sub-Nyquist sampling rate. By using traditional channel estimation methods, low-speed ADC sampling will result in low-resolution channel estimation. As a result, the overall system performance will be significantly degraded. To realize high-resolution channel estimation with low costs, it is necessary to develop high-resolution CCE techniques. Different from the methods proposed in our previous work [10,12,13], in this paper, we assume that the receiver is equipped with low-speed ADC as shown in Figure 3. On the basis of the low-speed ADC sampling system, we propose a high-resolution CCE method. Channel estimation problem is formulated for sub-Nyquist ADC sampling-based system, and high-resolution CCE method is proposed to estimate the sparse channel. In addition, to obtain robust channel estimator, we design subNyquist sampling rate-based training matrix to satisfy the restricted isometry property (RIP) [16] in the framework of compressive sensing (CS) [17,18].
1.3. Relations to other works In our previous works [10,12,13], sparse channel estimation methods have been proposed for broadband communication system using Nyquist rate sampling ADC at the receiver. Although the proposed methods can take advantage of the channel sparsity, high-speed ADC sampling is required at the receiver to satisfy the Nyquist sampling rate. It will pose a big challenge on the high-speed ADC design for industry. In addition, high-speed ADC is also too expensive for ordinary wireless communication. Unlike the previous methods, we consider sub-Nyquist sampling ADC at the receiver in this study. The benefit of the system is to relax the requirement of high-speed ADC sampling and therefore reduce the communication cost. In [19], CS-based high-resolution channel estimation method has been proposed for orthogonal frequency division modulation systems. The authors utilized low-speed ADC to reduce the sampling speed by using random convolution, and high-resolution channel estimation was achieved by iterative support detection algorithm [20]. Different from this work, we use consecutive piecewise integration to realize high-resolution CCE when the sub-Nyquist sampling rate ADC is used. Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
G. Gui, W. Peng and F. Adachi
Compressive channel estimation with low-speed ADC
1.4. Outline and notations
discrete channel vector consists of a maximal number of sampling taps with
The rest of this paper is organized as follows. Section 2 introduces the system model and problem formulation. Section 3 discusses CCE method from the sub-Nyquist rate sampling perspective. In Section 4, simulation results and discussion on the performance of the channel estimators are given. Finally, concluding remarks are presented in Section 5. In this paper, boldface lower case letters x denote vectors, boldface capital letters X denote signal matrices, and lower case letters xŒn and x.n/ represent the discrete-time signal and continuous-time signal, respectively. EŒ stands for the expectation operation. XT , X , and X denote the matrix transposition, conjugate, and conjugate transposition operations of X, respectively. Considering the signal vector x, kxk0 accounts its number of nonzero entries; kxk1 denotes L1 -norm and computes its absolute value; kxk2 is the Euclidean norm of x; and kxk1 denotes infinity norm and selects its maximal absolute value. C denotes complex-valued representation of signals.
L D dW max e C 1
at the Nyquist rate sampling period 1=W . Hence, the physical channel impulse response h.t / can be approximately by [21] h.t / D
2.1. Nyquist rate sampling-based system model
rŒn D
Z
nD0
(2)
max
h. /x.t /d C z.t /
r.t / D
h.l/ı .l l=W /
(5)
XL1 lD0
hŒlx Œn l C zŒn
(6)
Z
max
h. /fDAC .t C n=W /fADC .t /dt d
1 0
(7) with l D 0; 1; : : : ; L 1, and the discrete-time noise R1 zŒn D 1 z.t /fADC .n=W t /dt. From Equation (7), it can be found that the equivalent system channel vector h has a linear relationship with Nyquist sampling rate ADC. Assuming a fixed time-delay spread, the number of sampling channel taps increases with the transmission bandwidth, as shown in Figure 1.
0.7 0.6 Dominant channel taps
0.5
xŒnfDAC .t n=W /
where n=W is the nth up-sampling period and fDAC .t / D sinc.t / D sin.t /=t is the normalized sinc function. The waveform x.t / is transmitted over a frequency-selective fading channel h.t / with additive Gaussian noise interference z.t /; the receive continuous-time signal waveform is obtained as Z
1
hŒl D
(3)
Magnitude
XN 1
lD0
where the equivalent discrete-time channel impulse response
According to the Shannon sampling theorem, utilizing digital-to-analog converter (DAC) with impulse response fDAC .t /, which uniformly works at W hertz, the discretetime signal x can be converted into the continuous-time transmit waveform x.t / D
XL1
Here, we assume that the L-length discrete channel vector h D ŒhŒ0; h Œ1 ; : : : ; h ŒL 1T is supported by only K, (K L/ dominant channel taps, which is often termed as K-sparse multipath channel. A simple example is shown in Figure 4, where K D 5 denotes the number of dominant taps and L D 100 denotes the channel sampling length. Combining Equations (2-3), we obtain a discretetime channel that is described by the following relation between the discrete-time signals xŒn and rŒn
2. SYSTEM MODEL AND PROBLEM FORMULATION A sparse multipath broadband communication system is considered in this paper. We assume that the discrete transmit data sequence is xd D fxd Œn; n D 0; 1; : : : ; Nd 1g and the training sequence is x D fxŒn; nD0; 1; : : : ; N 1g. Hereby, the transmit data block is composed orthogonally of the data sequence xd and training sequence x. The focus of this study is channel estimation, and only the transmission of the training sequence is considered.
(4)
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0
where max denotes the maximum time-delay spread of the channel. According to the Shannon sampling theorem, Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
Figure 4. An example of sparse multipath channel with the overall sampling length 100 and the number of dominant channel taps is 5.
Compressive channel estimation with low-speed ADC
G. Gui, W. Peng and F. Adachi
If the number of sampling channel taps is L, Shannon sampling theorem suggests that the length of sampling on the received signal should be N (N L/. Even though ADC sampling speed have been increased, state-of-the-art architectures are not yet adequate for the emerging broadband communication systems [15,22–24]. Fortunately, wireless channel is often sparse and is supported by only a few dominant channel taps. Hence, only the dominant taps are necessary to be estimated. If the channel sparsity can be exploited, then the length of sampling on the received signal reduces to M O.K log.N =K// rather than N . Hence, in the next-generation broadband communication systems, CCE is one of the key technologies to reduce the burden of high-speed sampling ADC at the receiver.
we can obtain the matrix–vector form input–output system relation, given by y D QPXh C z D ‚h C z
where y DŒy Œ0 ; y Œ1 ; : : : ; yŒN 1T 2 C M 1 denotes the received signal vector, z D Œz Œ0 ; z Œ1 ; : : : zŒN 1T 2 C M 1 is an observed additive white Gaussian noise vector that is distributed as CN 0n2 IM ; h is a discrete-time channel vector; ‚ D QPX denotes a sub-Nyquist rate sampling training matrix with its column vector l D PN nD1 pn xnl qn , l D 1; 2; : : : ; L, xnl is the .nl/-th entry of X; X is the N L equivalent discrete circulant training matrix, 2
2.2. Sub-Nyquist rate sampling-based system model Different from the receiver with Nyquist sampling rate ADC, here, received signal waveform r.t / is sampled by low-speed ADC with impulse response fS_DAC .t / working at sub-Nyquist rate R, .R W /. As is shown in Figure 5, the components of the sub-Nyquist rate sampling ADC include a pseudorandom number (PN) generator, a mixer, a consecutive integrator, and a low-speed sampling ADC. When the received waveform r.t / is input to the subNyquist ADC, it is multiplied by the PN sequence p.t / and then segmented by consecutive integrator. Then, at the mth integral window, the output discrete-time signal is given by Z
mC1=R
y.t /dt
yŒm D R Z
m=R mC1=R m=R
Z
mC1=R
Z
max
h .t / x.t /p.t /ddt
DR m=R
Z
0
mC1=R
z.t /p.t /dt
CR
6 6 6 4
xŒL xŒL C 1 :: : xŒN C L 1
(8)
m=R
for m D 0; 1; : : : ; M 1, where M D RN =W denotes the length of sub-Nyquist rate sampling; if W D R, then the sub-Nyquist sampling length M equals to the Nyquist sampling length N . According to previous Equations (2–8),
::: ::: :::
xŒ2 xŒ3 :: : xŒN C 1
xŒ1 xŒ2 :: : xŒN
3 7 7 7 5
(10)
7 N N 52C
(11)
and 2 6 P D diagfpg D 4
3
p1 ::
: pN
is the random diagonal demodulation matrix whose diagonal entries p D Œp0 p1 ; : : : ; pN are generated by PN binary sequences .C1=1/ with equal probability; the matrix Q is given by 2
r.t /p.t /dt
DR
(9)
6 6 6 QD6 6 6 4
3
q1 ::
7 7 7 7 2 C N N 7 7 5
: qm ::
:
(12)
qM It is an equivalent sub-Nyquist sampling matrix. q 2 C 1N =M denotes each segmented N =M consecutive unit entries in mth row and starts in each column mN =M C 1 for each m D 0; 1; : : : ; M 1. In the next section, we focus on CCE methods for the sub-Nyquist rate sampling-based communication systems.
3. COMPRESSIVE CHANNEL ESTIMATION 3.1. Review of the compressive sensing theory Figure 5. The components of the sub-Nyquist rate sampling ADC include a PN generator, a mixer, a segmented integrator, and a low-speed sampling ADC.
The CS theory [17,18] states that a K-sparse channel vector h can be robustly estimated from Equation (10), where ‚ is equivalent sub-Nyquist matrix with M rows and L Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
G. Gui, W. Peng and F. Adachi
Compressive channel estimation with low-speed ADC
columns, M < L, and z is the additive noise. Mathematically, the optimal compressive channel estimator hopt can be achieved by hopt D arg min h
1 ky ‚hk22 C 0 khk0 2
According to Equation (17), if ‚ satisfies the RIP, then the most significant eigenvalue of any K K principal sub-matrix of .‚ ‚ I/ should be satisfied.
˚ˇ ˇ D E ˇ ‚ ‚ I ˇ ıK
(13)
where 0 is a regularization parameter, which balances the estimation error and channel sparsity. Unfortunately, Equation (13) is non-deterministic polynomial-time hard problem [18]. In other words, optimal compressive channel estimators are unlikely to be calculated efficiently even in noiseless environment. Fortunately, numerous practical alternative CS algorithms can acquire a suboptimal solution for the channel h if the ‚ satisfies the RIP [16] in Equation (10). For any K-sparse channel vector h in the noise observation model in Equation (13), if the training matrix ‚ satisfies
(18)
where jk‚kj D supjjK k.‚j /k and sup denotes ; m D 1; 2; : : : ; M supreme function. Assuming that m denotes the mth row of ‚, then the Gram matrix of ‚ can be expressed by M X
‚ ‚ D
m ˝ m
(19)
mD1
Following the definition of the RIP, we can get .1 ıK /khk22 k‚hk22 .1 C ıK / khk22
(14)
with high probability with a parameter ıK 2 .0; 1/, then accurate channel estimation can be obtained. It is worth noting that reasonable sub-Nyquist rate R should be chosen to ensure ‚ to satisfy RIP with a high probability. In other words, the sub-Nyquist sampling rate R should be appropriate for the following theorem so that ‚ satisfies RIP with high probability. 3.2. Restricted isometry property for equivalent training matrix For the sub-Nyquist sampling receiver, accurate channel estimation can be acquired if we can choose appropriate sampling rate R so that the equivalent training matrix ‚ satisfies RIP. The theorem of RIP is introduced next. Theorem 1. (The RIP for equivalent training matrix ‚ based on the sub-Nyquist rate sampling [22]): If the matrix ‚ satisfies RIP with constant ıK 2 .0; 1/ with a high probability 1 O.W1 /, then sub-Nyquist sampling rate R should be satisfied. 2 Klog6 W R C ıK
mD1
(20)
where f m m D 1; 2; : : : ; M g is an independent copy of fm m D 1; 2; : : : ; M g. In other words, each m m D 1; 2; : : : ; M is chosen randomly, and it can keep orthogonal to other vectors with high probability. For this aim, it is necessary to analyze the Gram matrix of ‚ D QPX. Let n and n0 be the column indices in Q, P, and X; it is easy to find that column vectors xn ; n D 1; 2; : : : ; N are orthogonal to each other. It can be obtained that
hm ; m0 i D
XN n;n0 D1
D ımm0 C
pn pn0 hqm ; qm0 i xnm xn0 m0
XN n¤n0
pn pn0 hqm ; qm0 i xnm xn0 m0 (21)
(15)
Proof . The RIP of the matrix ‚ has been given in Equation (15), which can also be rewritten as ˇ ˇ ˇ k‚hk2 khk2 ˇ ˇ 2 2ˇ ˇ ıK ˇ ˇ ˇ khk22
ˇ ˚ˇ D E ˇ‚ ‚ Iˇ ˇ ˇX ˇ M ˇ ˇ ˇ DE ˝ I m m ˇ mD1 ˇ ˇ ˇX ˇ M ˇ ˇ ˇ DE ˝ Ef ˝ g m m m m ˇ ˇ
(16)
where ımm0 denotes the Kronecker delta function, that is, ımm0 D 1 if m D m0 and ımm0 ¤ 0 if m¤ m0 . Hence, the Gram matrix of ‚ can be divided as ‚ ‚ D I C D
(22)
for khk0 K. It is not easy to find the relationship between RIP and ‚ in Equation (16). For this reason, we resort to its equivalent equation as
P where dmm0 D N n¤n0 pn pn0 hqm ; qm0 i xnm xn0 m0 . It is clear that EfDg D 0 so that
ˇ ˇ ˇ h .‚ ‚ I/h ˇ ˇ ıK ˇ ˇ ˇ h h
˚ E ‚ ‚ D I
(17)
Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
(23)
Compressive channel estimation with low-speed ADC
G. Gui, W. Peng and F. Adachi
According to the symmetrization theorem in [22], Equation (20) can be rewritten as ˇ ˇX ˇ ˇ M ˇ ˇ p ˝ 2E m ˇ ˇ mD1 m m ( ˇX p ˇ M ˇ 2E C1 B K log2 W ˇ
mD1
p 2 6C1 s s
s
Input
An N L complex training matrix X An M N sub-Nyquist sampling matrix Q An N N diagonal modulation matrix P An M -dimensional received vector y p A regularized parameter D 2 log K
Run
cvx_begin variable h.L/ minimize fky ‚hk1 C khk1 g cvx_end
Output
An M -dimensional channel estimator hDS
ˇ1=2 ) ˇ m ˝ m ˇˇ
(ˇ ˇ1=2 ) ˇXM ˇ K log5 W ˇ m ˝ m ˇˇ E ˇ mD1 R
C2 K log5 W 2 1=2 C1 R C2 K log5 W R
(24)
where B D max fxmn g is a random entry of X. To m;n
guarantee that ‚ satisfies RIP, s
Table I. Sub-Nyquist rate sampling-based compressive channel estimation Dantzig selector.
C2 K log5 W ıK R
( (25)
if R C2 ı 2 K log5 W , then ‚ satisfies RIP with the probability ˇ ˚ˇ P ˇ‚ ‚ Iˇ < ıK D 1 O W 1
The tap position set of channel h is defined as , and we assume the position set of its dominant taps denoted by K is known. By using the prior information, the lower bound of channel estimator hS_ADC can be given by
(26)
3.3. Compressive channel estimation We formulate the sub-Nyquist rate sampling-based channel estimation as a CS problem [17,18] and also name the estimation methods as CCE. Sparse channel estimation methods [6–12] have been intensively studied in recent years. However, all of the proposed methods are based on the Nyquist rate sampling. The estimation methods can be classified by two types: one is mixed-norm-based convex relation algorithm, for example, least absolute shrinkage and selection operator [25] and Dantzig selector (DS) [26], and the other is greedy iterative algorithm, for example, orthogonal matching pursuit [27] and compressive sampling matching pursuit [28]. Considering the sub-Nyquist sampling, CCE method is implemented by DS algorithm, which is termed as CCE-DS. The flowchart of CCE-DS is shown in Table I. 3.4. Lower bound of compressive channel estimator To evaluate the performance of the proposed methods, we give lower bound on the basis of sub-Nyquist ADC sampling. Consider the sub-Nyquist ADC sampling, which is shown in Figure 3; the equivalent training matrix is ‚ 2 C M L , and the discrete received signal is y 2 C M 1 .
hS_ADC D
arg minK r ‚ K hK 2 ;
K
0;
= K (27) where hK contains the K dominant channel taps of h and ‚ K is the sub-matrix constructed from the K columns of X. In the following, the lower bound of channel estimator hS_ADC is obtained by average mean square error (MSE), given by o 1 n H ‚ ‚ z E kh hS_ADC k22 D E ‚ H K K K 1 H ‚H D Trace ‚ H K ‚ K K zz 1 ‚ K ‚ H K ‚ K 1 ‚ D n2 Trace ‚ H K K
2K2 o n n Trace ‚ H K ‚ K
D n2 K=N where Trace
n
‚H K ‚ K
o
(28)
D N =K.
4. NUMERICAL SIMULATIONS In this section, we will compare the performance of the proposed estimator with least square (LS)-based linear estimator by numerical results on the basis of 1000 independent Monte Carlo runs. The broadband bandwidth is set to be W =2 D 100 MHz and channel delay spread max D 0:5 s. Hereby, the sampling length of channel vector h is set to be L D 100, and the number of dominant channel taps of h is K D 4 where the positions of Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
Compressive channel estimation with low-speed ADC
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Average MSE (overall channel taps)
G. Gui, W. Peng and F. Adachi
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dominant taps are generated following Gaussian distribution and subject to khk22 D 1. The Nyquist sampling rate of high-speed ADC is W D 200 MHz, and sub-Nyquist sampling rate of low-speed ADC is R D ˛W , where ˛ is a sub-Nyquist sampling factor. In this simulation, we consider the sub-Nyquist sampling factor to be ˛ D 0:4; 0:6, and 0:8. The length of training sequence is N D 120. The received signal-to-noise ratio is defined as 10 log.P =n2 /. Channel estimators hO are evaluated by average MSE, which is defined by
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(29)
where h and hO denote the channel vector and its estimator, respectively, and L is the sampling length. In Figure 6, we compare the average MSE performance of the proposed channel estimator and the LS channel estimator on the overall channel taps. It is found that the sub-Nyquist sampling rate-based compressive channel estimators are better than Nyquist rate sampling-based LS channel estimator. For the sparse multipath channel, the proposed method can estimate the dominant channel taps robustly, whereas the non-dominant channel taps are forced to zeros. Unlike the proposed method, LS-based channel estimator has to estimate all the channel taps uniformly even if the channel taps are equal to zero. It is worth mentioning that the LS channel estimator cannot work well on sub-Nyquist rate sampling, as shown in the purple curve in Figure 6; when sub-Nyquist rate R D 0:8 W is used for LS channel estimator, the average MSE performance has been significantly degraded. In Figure 7, the comparison of average MSE performance is taken on dominant channel taps. It is found that when the sub-Nyquist sampling rate R D 0:8 W, the proposed channel estimator can achieve the same average MSE performance as Nyquist sampling rate LS channel estimator. Even for the cases when R D 0:4 W and R D 0:6 W, the performances of the proposed estimator are also Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
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Proposed (DS, K=2) Proposed (DS, K=4) Proposed (DS, K=6) Proposed (DS, K=8) LS (R=0.4W) LS (R=W) Lower bound (known channel, R=0.4W)
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SNR (dB) Figure 8. Average MSE performance of overall channel taps.
Average MSE (dominant channel taps)
2 Average MSE hO D E h hO =L
Figure 7. Average MSE performance of dominant channel taps.
Average MSE (overall channel taps)
Figure 6. Average MSE performance of overall channel taps.
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close to LS-based one. From Figures 6 and 7, it can be concluded that the proposed channel estimator can achieve almost the same high-resolution channel estimation as the Nyquist sampling rate LS channel estimator.
Compressive channel estimation with low-speed ADC
To further confirm the advantage of the proposed method, we vary the channel sparsity, and the average MSE performances are evaluated and shown in Figures 8 and 9. The number of nonzero channel taps is set to be K D 2; 4; 6; and 8, respectively. And the sub-Nyquist sampling rate is set to be R D 0:4 W. Figure 8 depicts the average MSE performance of the overall channel taps. It is observed that sub-Nyquist sampling-based CCE method can achieve better estimation than Nyquist sampling-based LS method. In addition, as shown in the blue curves in Figure 8, the proposed method can robustly estimate the channels with different channel sparsity under the sub-Nyquist sampling rate. It is worth mentioning that the sparser the channel is, the better estimation performance can be achieved by the proposed method. However, the LS channel estimator can obtain robust performance only under the Nyquist sampling rate; when sub-Nyquist sampling rate is used, the performance is significantly degraded. Figure 9 shows the average MSE performance of the dominant channel taps. It is shown that the proposed subNyquist sampling rate-based channel estimator can achieve close performance as the Nyquist sampling rate-based LS channel estimator.
G. Gui, W. Peng and F. Adachi
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5. CONCLUSION Broadband transmission poses a big challenge on ADC sampling capability in the next-generation wireless communication systems. To avoid the high-speed ADC sampling at receiver, we consider low-speed sampling ADC at the receiver in this paper. The channel estimation has been formulated as a CS problem, and a CCE method using DS algorithm has been proposed. By comparing with traditional Nyquist sampling rate LS channel estimation, the proposed method can reduce sampling rate while achieving similarly good estimation performance. The effectiveness of the proposed method has been verified by the simulation results.
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ACKNOWLEDGEMENTS The authors would like to thank the anonymous reviewers for their kind and constructive comments, which helped much to improve the quality of the paper. This work was supported by the Japan Society for the Promotion of Science postdoctoral fellowship.
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REFERENCES 1. Adachi F, Kudoh E. New direction of broadband wireless technology. Wireless Communications and Mobile Computing 2007; 7(8): 969–983. 2. He SB, Chen JM, Sun YX, Yau D, Yip NK. On optimal information capture by energy-constrained mobile
13.
sensors. IEEE Transactions on Vehicular Technology 2010; 59(5): 2472–2484. Chen JM, Li JK, He SB, Sun YX, Chen HH. Energyefficient coverage based on probabilistic sensing model in wireless sensor networks. IEEE Communication Letters 2010; 14(9): 833–835. Chen H, Wang G, Wang Z, So H, Poor H. Nonline-of-sight node localization based on semi-definite programming in wireless sensor networks. IEEE Transactions on Wireless Communications 2012; 11(1): 108–116. Shi Q, He C, Chen H, Jiang L. Distributed wireless sensor network localization via sequential greedy optimization algorithm. IEEE Transactions on Signal Processing 2010; 58(6): 3328–3340. Cotter SF, Rao BD. Sparse channel estimation via matching pursuit with application to equalization. IEEE Transactions on Communications 2002; 50(3): 374–377. Carbonelli C, Vedantam S, Mitra U. Sparse channel estimation with zero tap detection. IEEE Transactions on Wireless Communications 2007; 6(5): 1743–1753. Bajwa UW, Haupt J, Raz G, Nowak R. Compressed channel sensing, In 42th Annual Conference on Information Sciences and Systems (CISS), Princeton, NJ, USA, March 19–21, 2008; 5–10. Taubock G, Hlawatsch F. A compressed sensing technique for OFDM channel estimation in mobile environments: Exploiting channel sparsity for reducing pilots, In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, Nevada, USA, March 31–April 4, 2008; 1–5. Gui G, Wan Q, Huang AM, Chen ZX. Sparse multipath channel estimation using Dantzig selector algorithm, In The 12th International Symposium on Wireless Personal Multimedia Communications (WPMC), Sendai, Japan, September 7–10, 2009; 1–5. Berger CR, Zhou S, Preisig JC, Willett P. Sparse channel estimation for multicarrier underwater acoustic communication: from subspace methods to compressed sensing. IEEE Transactions on Signal Processing 2010; 58(3): 1708–1721. Gui G, Wan Q, Peng W, Adachi F. Sparse multipath channel estimation using compressive sampling matching pursuit algorithm, In 7th IEEE Vehicular Technology Society Asia Pacific Wireless Communication Symposium (APWCS), Kaohsiung, Taiwan, May 20–21, 2010; 1–5. Gui G, Peng W, Adachi F. Improved adaptive sparse channel estimation based on the least mean square algorithm, In IEEE Wireless Communications and Networking Conference (WCNC), Shanghai, China, April 7–10, 2013; 1–5.
Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
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14. Marks RJ. Introduction to Shannon Sampling and Interpolation Theory. Spinger-Verlag: Berlin, Germany, 1991. 15. Raychaudhuri D, Mandayam NB. Frontiers of wireless and mobile communications. Proceedings of the IEEE 2012; 100(4): 824–840. 16. Candes EJ. The restricted isometry property and its implications for compressed sensing. Compte Rendus de l’Academie des Sciences 2008; Serie I(346): 589–592. 17. Candes E, Romberg J, Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transaction on Information Theory 2006; 52(2): 489–509. 18. Donoho DL. Compressed sensing. IEEE Transactions on Information Theory 2006; 52(4): 1289–1306. 19. Meng J, Yin WT, Li YY, Nguyen NT. Compressive sensing based high-resolution channel estimation for OFDM system. IEEE Journal of Selected Topics in Signal Processing 2012; 6(1): 15–25. 20. Wang Y, Yin WT. Sparse signal reconstruction via iterative support detection. SIAM Journal on Imaging Sciences 2010; 3(3): 462–491. 21. Proakis JG. Digital Communications, 4th edition. New York, NY: McGraw-Hill, 2001. 22. Tropp JA, Laska JN, Duarte MF, Romberg JK, Baraniuk RG. Beyond Nyquist: efficient sampling of sparse bandlimited signals. IEEE Transaction on Information Theory 2010; 56(1): 520–544. 23. Mishali M, Eldar YC. From theory to practice: subNyquist sampling of sparse wideband analog signals. IEEE Journal of Selected Topics in Signal Processing 2010; 4(2): 375–391. 24. Chen X, Yu Z, Hoyos S, Sadler BM, Silva-Martinez J. A sub-Nyquist rate sampling receiver exploiting compressive sensing. IEEE Transactions on Circuits and Systems I: Regular Papers 2011; 58(3): 507–520. 25. Ribshirani T. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society (B) 1996; 58(1): 267–288. 26. Candes EJ, Tao T. Rejoinder: the Dantzig selector: statistical estimation when p is much larger than n. Annals of Statistics 2007; 35(6): 2392–240. 27. Tropp JA, Gilbert AC. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transaction on Information Theory 2007; 5(12): 4655–4666. 28. Needell D, Tropp JA. CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis 2008; 26(3): 301–321.
Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
Compressive channel estimation with low-speed ADC
AUTHORS’ BIOGRAPHIES Guan Gui received the Dr.Eng. degree in information and communication engineering from University of Electronic Science and Technology of China, Chengdu, China, in 2011. From March 2009 to July 2011, he was selected as outstanding doctor training candidate by the University of Electronic Science and Technology of China. From October 2009 to March 2012, with the financial support from the China scholarship council and the global center of education of Tohoku University, he joined the wireless signal processing and network laboratory (Professor Adachi’s laboratory), Department of Communication Engineering, Graduate School of Engineering, Tohoku University as a research assistant and postdoctoral research fellow, respectively. Since September 2012, he is supported by Japan society for the Promotion of Science fellowship as postdoctoral research fellow at same laboratory. His research interests are adaptive system identification, compressive sensing, sparse dictionary designing, channel estimation, and advanced wireless techniques. He is member of IEEE and IEICE. Wei Peng received her BS and MS degrees in electrical engineering from Wuhan University, Wuhan, China, in 2000 and 2003, respectively. She received the Dr.Eng. degree in electrical and electronic engineering from the University of Hong Kong, Hong Kong, in 2007. Since December 2007, she jointed Tohoku University, and worked as an assistant professor. Currently she is with the department of Huazhong University of Science and Technology, Wuhan, China, as an associate professor. Her research interests are in multiple antenna technology, cellular system, and distributed antenna network. She is an IEEE senior member. Fumiyuki Adachi received the BS and Dr.Eng. degrees in electrical engineering from Tohoku University, Sendai, Japan, in 1973 and 1984, respectively. In April 1973, he joined the Electrical Communications Laboratories of Nippon Telegraph & Telephone Corporation (now NTT) and conducted various types of research related to digital cellular mobile communications. From July 1992 to December 1999, he was with NTT Mobile Communications Network, Inc. (now NTT DoCoMo, Inc.), where he led a research group on wideband/broadband CDMA wireless access for IMT2000 and beyond. Since January 2000, he has been with
Compressive channel estimation with low-speed ADC
Tohoku University, Sendai, Japan, where he is a professor of Communication Engineering at the Graduate School of Engineering. He is currently engaged in research of gigabit wireless communication technology with a data rate above one gigabit per second, with the aim to realize the next-generation frequency and energy-efficient broadband mobile communication systems. He has been serving as the Institute of Electrical and Electronics Engineers VTS Distinguished Lecturer since 2011.From October 1984 to September 1985, he was a United Kingdom SERC Visiting Research Fellow in the Department of Electrical
G. Gui, W. Peng and F. Adachi
Engineering and Electronics at Liverpool University. He is an IEICE Fellow and was a co-recipient of the IEICE Transactions best paper of the year award 1996, 1998, and 2009 and also a recipient of Achievement Award 2003. He is an IEEE Fellow and was a co-recipient of the IEEE Vehicular Technology Transactions best paper of the year award 1980 and again 1990 and also a recipient of Avant Garde award 2000. He was a recipient of Thomson Scientific Research Front Award 2004, Ericsson Telecommunications Award 2008, and Telecom System Technology Award 2010.
Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
RESEARCH ARTICLE
Sub-Nyquist rate ADC sampling-based compressive channel estimation Guan Gui1 , Wei Peng1,2* and Fumiyuki Adachi1 1 2
Tohoku University, Department of Communication Engineering Huazhong University of Science and Technology, Department of Electronics and Information Engineering
ABSTRACT To realize high-speed communication, broadband transmission has become an indispensable technique in the nextgeneration wireless communication systems. Broadband channel is often characterized by the sparse multipath channel model, and significant taps are widely separated in time, and thereby, a large delay spread exists. Accurate channel state information is required for coherent detection. Traditionally, accurate channel estimation can be achieved by sampling the received signal with large delay spread by analog-to-digital converter (ADC) at Nyquist rate and then estimate all of channel taps. However, as the transmission bandwidth increases, the demands of the Nyquist sampling rate already exceed the capabilities of current ADC. In addition, the high-speed ADC is very expensive for ordinary wireless communication. In this paper, we present a novel receiver, which utilizes a sub-Nyquist ADC that samples at much lower rate than the Nyquist one. On the basis of the sampling scheme, we propose a compressive channel estimation method using Dantzig selector algorithm. By comparing with the traditional least square channel estimation, our proposed method not only achieves robust channel estimation but also reduces the cost because low-speed ADC is much cheaper than high-speed one. Computer simulations confirm the effectiveness of our proposed method. Copyright © 2013 John Wiley & Sons, Ltd. KEYWORDS compressive channel estimation; analog-to-digital converter (ADC); sub-Nyquist rate sampling; compressive sensing *Correspondence Wei Peng, Huazhong University of Science and Technology, Department of Electronics and Information Engineering,Luoyu Road 1037,Hongshan District, Wuhan, 430074,China. E-mail: [email protected]
1. INTRODUCTION 1.1. Background and motivation With the increasing number of wireless subscribers, various portable wireless devices, for example, smartphones and laptops, have generated increasing massive data traffic. The demand for high-speed data services is getting more difficult to be satisfied. Broadband transmission is an indispensable technique in the next-generation communication systems [1] and sensor networks [2–5]. However, the broadband signal is susceptible to interference caused by frequency-selective fading. Consequently, accurate channel estimation is required at the receiver for coherent detection. Because the broadband channel is often described by the sparse channel model, sparse channel estimation methods have been proposed to take the advantage of channel sparsity [6–13]. By exploiting the sparse prior
Copyright © 2013 John Wiley & Sons, Ltd.
information, channel estimation performance or spectral efficiency can be improved. Nyquist rate analog-to-digital converter (ADC) sampling system based on traditional sparse channel estimation is shown in Figure 1. As the data rate increases, such a system poses two challenges for the next-generation broadband wireless communication systems. The first challenge is that the requirement of Nyquist sampling speed exceeds the capability of current ADC. Besides, high-speed ADC device becomes very expensive when the sampling rate increases. The second challenge is that Nyquist samplingbased system will reduce the spectrum efficiency because of the large number of training sequence used for channel estimation. Therefore, it is necessary to develop a novel technique to relax the requirement on high-speed ADC sampling in the broadband communication systems. Shannon sampling theorem [14] is one of the fundamental theorems of modern signal processing. Given a
Compressive channel estimation with low-speed ADC
G. Gui, W. Peng and F. Adachi
Figure 1. Communication system using Nyquist sampling rate ADC.
Figure 3. Broadband communication system using sub-Nyquist sampling rate ADC.
continuous signal r.t /, t 2 Œ0; T / whose highest frequency is less than W =2 hertz, Shannon theorem suggests that the signal be sampled uniformly at a rate of W hertz. The signal can be recovered from the samples, given by
1.2. Main contribution
r.t / D
X n2
r
n sinc.W t n/; t 2 Œ0; T / W
(1)
Number of channel sampling taps
However, this well-known approach becomes impractical when the transmission rate or signal bandwidth is large because it is challenging to build a sampling device that operates at a sufficiently high speed. The demands of many modern applications already exceed the capabilities of current technology [15]. Even though recent developments in ADC have increased the sampling rate, state-of-the-art architectures are not yet adequate for the emerging applications. According to the Shannon sampling theorem [14], the number of taps in channel sampling has a linear relation with the transmission bandwidth. And the number of channel sampling taps easily reaches hundreds when its transmission bandwidth becomes large. A simple example of the number of channel sampling taps is depicted in Figure 2.
2
10
1
10
1 2 3
0
10
0
20
40
60
80
=1 s =0.5 s =0.1 s
100
Transmission bandwidth (MHz) Figure 2. Relation between the number of channel sampling taps with Nyquist rate sampling and transmission bandwidth at different time delay spreads: 1 D 1 s, 2 D 5 s, and 3 D 0:1 s.
In this paper, we investigate compressive channel estimation (CCE) problem on the basis of low-speed ADC working at sub-Nyquist sampling rate. By using traditional channel estimation methods, low-speed ADC sampling will result in low-resolution channel estimation. As a result, the overall system performance will be significantly degraded. To realize high-resolution channel estimation with low costs, it is necessary to develop high-resolution CCE techniques. Different from the methods proposed in our previous work [10,12,13], in this paper, we assume that the receiver is equipped with low-speed ADC as shown in Figure 3. On the basis of the low-speed ADC sampling system, we propose a high-resolution CCE method. Channel estimation problem is formulated for sub-Nyquist ADC sampling-based system, and high-resolution CCE method is proposed to estimate the sparse channel. In addition, to obtain robust channel estimator, we design subNyquist sampling rate-based training matrix to satisfy the restricted isometry property (RIP) [16] in the framework of compressive sensing (CS) [17,18].
1.3. Relations to other works In our previous works [10,12,13], sparse channel estimation methods have been proposed for broadband communication system using Nyquist rate sampling ADC at the receiver. Although the proposed methods can take advantage of the channel sparsity, high-speed ADC sampling is required at the receiver to satisfy the Nyquist sampling rate. It will pose a big challenge on the high-speed ADC design for industry. In addition, high-speed ADC is also too expensive for ordinary wireless communication. Unlike the previous methods, we consider sub-Nyquist sampling ADC at the receiver in this study. The benefit of the system is to relax the requirement of high-speed ADC sampling and therefore reduce the communication cost. In [19], CS-based high-resolution channel estimation method has been proposed for orthogonal frequency division modulation systems. The authors utilized low-speed ADC to reduce the sampling speed by using random convolution, and high-resolution channel estimation was achieved by iterative support detection algorithm [20]. Different from this work, we use consecutive piecewise integration to realize high-resolution CCE when the sub-Nyquist sampling rate ADC is used. Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
G. Gui, W. Peng and F. Adachi
Compressive channel estimation with low-speed ADC
1.4. Outline and notations
discrete channel vector consists of a maximal number of sampling taps with
The rest of this paper is organized as follows. Section 2 introduces the system model and problem formulation. Section 3 discusses CCE method from the sub-Nyquist rate sampling perspective. In Section 4, simulation results and discussion on the performance of the channel estimators are given. Finally, concluding remarks are presented in Section 5. In this paper, boldface lower case letters x denote vectors, boldface capital letters X denote signal matrices, and lower case letters xŒn and x.n/ represent the discrete-time signal and continuous-time signal, respectively. EŒ stands for the expectation operation. XT , X , and X denote the matrix transposition, conjugate, and conjugate transposition operations of X, respectively. Considering the signal vector x, kxk0 accounts its number of nonzero entries; kxk1 denotes L1 -norm and computes its absolute value; kxk2 is the Euclidean norm of x; and kxk1 denotes infinity norm and selects its maximal absolute value. C denotes complex-valued representation of signals.
L D dW max e C 1
at the Nyquist rate sampling period 1=W . Hence, the physical channel impulse response h.t / can be approximately by [21] h.t / D
2.1. Nyquist rate sampling-based system model
rŒn D
Z
nD0
(2)
max
h. /x.t /d C z.t /
r.t / D
h.l/ı .l l=W /
(5)
XL1 lD0
hŒlx Œn l C zŒn
(6)
Z
max
h. /fDAC .t C n=W /fADC .t /dt d
1 0
(7) with l D 0; 1; : : : ; L 1, and the discrete-time noise R1 zŒn D 1 z.t /fADC .n=W t /dt. From Equation (7), it can be found that the equivalent system channel vector h has a linear relationship with Nyquist sampling rate ADC. Assuming a fixed time-delay spread, the number of sampling channel taps increases with the transmission bandwidth, as shown in Figure 1.
0.7 0.6 Dominant channel taps
0.5
xŒnfDAC .t n=W /
where n=W is the nth up-sampling period and fDAC .t / D sinc.t / D sin.t /=t is the normalized sinc function. The waveform x.t / is transmitted over a frequency-selective fading channel h.t / with additive Gaussian noise interference z.t /; the receive continuous-time signal waveform is obtained as Z
1
hŒl D
(3)
Magnitude
XN 1
lD0
where the equivalent discrete-time channel impulse response
According to the Shannon sampling theorem, utilizing digital-to-analog converter (DAC) with impulse response fDAC .t /, which uniformly works at W hertz, the discretetime signal x can be converted into the continuous-time transmit waveform x.t / D
XL1
Here, we assume that the L-length discrete channel vector h D ŒhŒ0; h Œ1 ; : : : ; h ŒL 1T is supported by only K, (K L/ dominant channel taps, which is often termed as K-sparse multipath channel. A simple example is shown in Figure 4, where K D 5 denotes the number of dominant taps and L D 100 denotes the channel sampling length. Combining Equations (2-3), we obtain a discretetime channel that is described by the following relation between the discrete-time signals xŒn and rŒn
2. SYSTEM MODEL AND PROBLEM FORMULATION A sparse multipath broadband communication system is considered in this paper. We assume that the discrete transmit data sequence is xd D fxd Œn; n D 0; 1; : : : ; Nd 1g and the training sequence is x D fxŒn; nD0; 1; : : : ; N 1g. Hereby, the transmit data block is composed orthogonally of the data sequence xd and training sequence x. The focus of this study is channel estimation, and only the transmission of the training sequence is considered.
(4)
0.4 0.3 0.2 0.1 0
0
20
40
60
80
100
Sampling length of sparse channel
0
where max denotes the maximum time-delay spread of the channel. According to the Shannon sampling theorem, Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
Figure 4. An example of sparse multipath channel with the overall sampling length 100 and the number of dominant channel taps is 5.
Compressive channel estimation with low-speed ADC
G. Gui, W. Peng and F. Adachi
If the number of sampling channel taps is L, Shannon sampling theorem suggests that the length of sampling on the received signal should be N (N L/. Even though ADC sampling speed have been increased, state-of-the-art architectures are not yet adequate for the emerging broadband communication systems [15,22–24]. Fortunately, wireless channel is often sparse and is supported by only a few dominant channel taps. Hence, only the dominant taps are necessary to be estimated. If the channel sparsity can be exploited, then the length of sampling on the received signal reduces to M O.K log.N =K// rather than N . Hence, in the next-generation broadband communication systems, CCE is one of the key technologies to reduce the burden of high-speed sampling ADC at the receiver.
we can obtain the matrix–vector form input–output system relation, given by y D QPXh C z D ‚h C z
where y DŒy Œ0 ; y Œ1 ; : : : ; yŒN 1T 2 C M 1 denotes the received signal vector, z D Œz Œ0 ; z Œ1 ; : : : zŒN 1T 2 C M 1 is an observed additive white Gaussian noise vector that is distributed as CN 0n2 IM ; h is a discrete-time channel vector; ‚ D QPX denotes a sub-Nyquist rate sampling training matrix with its column vector l D PN nD1 pn xnl qn , l D 1; 2; : : : ; L, xnl is the .nl/-th entry of X; X is the N L equivalent discrete circulant training matrix, 2
2.2. Sub-Nyquist rate sampling-based system model Different from the receiver with Nyquist sampling rate ADC, here, received signal waveform r.t / is sampled by low-speed ADC with impulse response fS_DAC .t / working at sub-Nyquist rate R, .R W /. As is shown in Figure 5, the components of the sub-Nyquist rate sampling ADC include a pseudorandom number (PN) generator, a mixer, a consecutive integrator, and a low-speed sampling ADC. When the received waveform r.t / is input to the subNyquist ADC, it is multiplied by the PN sequence p.t / and then segmented by consecutive integrator. Then, at the mth integral window, the output discrete-time signal is given by Z
mC1=R
y.t /dt
yŒm D R Z
m=R mC1=R m=R
Z
mC1=R
Z
max
h .t / x.t /p.t /ddt
DR m=R
Z
0
mC1=R
z.t /p.t /dt
CR
6 6 6 4
xŒL xŒL C 1 :: : xŒN C L 1
(8)
m=R
for m D 0; 1; : : : ; M 1, where M D RN =W denotes the length of sub-Nyquist rate sampling; if W D R, then the sub-Nyquist sampling length M equals to the Nyquist sampling length N . According to previous Equations (2–8),
::: ::: :::
xŒ2 xŒ3 :: : xŒN C 1
xŒ1 xŒ2 :: : xŒN
3 7 7 7 5
(10)
7 N N 52C
(11)
and 2 6 P D diagfpg D 4
3
p1 ::
: pN
is the random diagonal demodulation matrix whose diagonal entries p D Œp0 p1 ; : : : ; pN are generated by PN binary sequences .C1=1/ with equal probability; the matrix Q is given by 2
r.t /p.t /dt
DR
(9)
6 6 6 QD6 6 6 4
3
q1 ::
7 7 7 7 2 C N N 7 7 5
: qm ::
:
(12)
qM It is an equivalent sub-Nyquist sampling matrix. q 2 C 1N =M denotes each segmented N =M consecutive unit entries in mth row and starts in each column mN =M C 1 for each m D 0; 1; : : : ; M 1. In the next section, we focus on CCE methods for the sub-Nyquist rate sampling-based communication systems.
3. COMPRESSIVE CHANNEL ESTIMATION 3.1. Review of the compressive sensing theory Figure 5. The components of the sub-Nyquist rate sampling ADC include a PN generator, a mixer, a segmented integrator, and a low-speed sampling ADC.
The CS theory [17,18] states that a K-sparse channel vector h can be robustly estimated from Equation (10), where ‚ is equivalent sub-Nyquist matrix with M rows and L Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
G. Gui, W. Peng and F. Adachi
Compressive channel estimation with low-speed ADC
columns, M < L, and z is the additive noise. Mathematically, the optimal compressive channel estimator hopt can be achieved by hopt D arg min h
1 ky ‚hk22 C 0 khk0 2
According to Equation (17), if ‚ satisfies the RIP, then the most significant eigenvalue of any K K principal sub-matrix of .‚ ‚ I/ should be satisfied.
˚ˇ ˇ D E ˇ ‚ ‚ I ˇ ıK
(13)
where 0 is a regularization parameter, which balances the estimation error and channel sparsity. Unfortunately, Equation (13) is non-deterministic polynomial-time hard problem [18]. In other words, optimal compressive channel estimators are unlikely to be calculated efficiently even in noiseless environment. Fortunately, numerous practical alternative CS algorithms can acquire a suboptimal solution for the channel h if the ‚ satisfies the RIP [16] in Equation (10). For any K-sparse channel vector h in the noise observation model in Equation (13), if the training matrix ‚ satisfies
(18)
where jk‚kj D supjjK k.‚j /k and sup denotes ; m D 1; 2; : : : ; M supreme function. Assuming that m denotes the mth row of ‚, then the Gram matrix of ‚ can be expressed by M X
‚ ‚ D
m ˝ m
(19)
mD1
Following the definition of the RIP, we can get .1 ıK /khk22 k‚hk22 .1 C ıK / khk22
(14)
with high probability with a parameter ıK 2 .0; 1/, then accurate channel estimation can be obtained. It is worth noting that reasonable sub-Nyquist rate R should be chosen to ensure ‚ to satisfy RIP with a high probability. In other words, the sub-Nyquist sampling rate R should be appropriate for the following theorem so that ‚ satisfies RIP with high probability. 3.2. Restricted isometry property for equivalent training matrix For the sub-Nyquist sampling receiver, accurate channel estimation can be acquired if we can choose appropriate sampling rate R so that the equivalent training matrix ‚ satisfies RIP. The theorem of RIP is introduced next. Theorem 1. (The RIP for equivalent training matrix ‚ based on the sub-Nyquist rate sampling [22]): If the matrix ‚ satisfies RIP with constant ıK 2 .0; 1/ with a high probability 1 O.W1 /, then sub-Nyquist sampling rate R should be satisfied. 2 Klog6 W R C ıK
mD1
(20)
where f m m D 1; 2; : : : ; M g is an independent copy of fm m D 1; 2; : : : ; M g. In other words, each m m D 1; 2; : : : ; M is chosen randomly, and it can keep orthogonal to other vectors with high probability. For this aim, it is necessary to analyze the Gram matrix of ‚ D QPX. Let n and n0 be the column indices in Q, P, and X; it is easy to find that column vectors xn ; n D 1; 2; : : : ; N are orthogonal to each other. It can be obtained that
hm ; m0 i D
XN n;n0 D1
D ımm0 C
pn pn0 hqm ; qm0 i xnm xn0 m0
XN n¤n0
pn pn0 hqm ; qm0 i xnm xn0 m0 (21)
(15)
Proof . The RIP of the matrix ‚ has been given in Equation (15), which can also be rewritten as ˇ ˇ ˇ k‚hk2 khk2 ˇ ˇ 2 2ˇ ˇ ıK ˇ ˇ ˇ khk22
ˇ ˚ˇ D E ˇ‚ ‚ Iˇ ˇ ˇX ˇ M ˇ ˇ ˇ DE ˝ I m m ˇ mD1 ˇ ˇ ˇX ˇ M ˇ ˇ ˇ DE ˝ Ef ˝ g m m m m ˇ ˇ
(16)
where ımm0 denotes the Kronecker delta function, that is, ımm0 D 1 if m D m0 and ımm0 ¤ 0 if m¤ m0 . Hence, the Gram matrix of ‚ can be divided as ‚ ‚ D I C D
(22)
for khk0 K. It is not easy to find the relationship between RIP and ‚ in Equation (16). For this reason, we resort to its equivalent equation as
P where dmm0 D N n¤n0 pn pn0 hqm ; qm0 i xnm xn0 m0 . It is clear that EfDg D 0 so that
ˇ ˇ ˇ h .‚ ‚ I/h ˇ ˇ ıK ˇ ˇ ˇ h h
˚ E ‚ ‚ D I
(17)
Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
(23)
Compressive channel estimation with low-speed ADC
G. Gui, W. Peng and F. Adachi
According to the symmetrization theorem in [22], Equation (20) can be rewritten as ˇ ˇX ˇ ˇ M ˇ ˇ p ˝ 2E m ˇ ˇ mD1 m m ( ˇX p ˇ M ˇ 2E C1 B K log2 W ˇ
mD1
p 2 6C1 s s
s
Input
An N L complex training matrix X An M N sub-Nyquist sampling matrix Q An N N diagonal modulation matrix P An M -dimensional received vector y p A regularized parameter D 2 log K
Run
cvx_begin variable h.L/ minimize fky ‚hk1 C khk1 g cvx_end
Output
An M -dimensional channel estimator hDS
ˇ1=2 ) ˇ m ˝ m ˇˇ
(ˇ ˇ1=2 ) ˇXM ˇ K log5 W ˇ m ˝ m ˇˇ E ˇ mD1 R
C2 K log5 W 2 1=2 C1 R C2 K log5 W R
(24)
where B D max fxmn g is a random entry of X. To m;n
guarantee that ‚ satisfies RIP, s
Table I. Sub-Nyquist rate sampling-based compressive channel estimation Dantzig selector.
C2 K log5 W ıK R
( (25)
if R C2 ı 2 K log5 W , then ‚ satisfies RIP with the probability ˇ ˚ˇ P ˇ‚ ‚ Iˇ < ıK D 1 O W 1
The tap position set of channel h is defined as , and we assume the position set of its dominant taps denoted by K is known. By using the prior information, the lower bound of channel estimator hS_ADC can be given by
(26)
3.3. Compressive channel estimation We formulate the sub-Nyquist rate sampling-based channel estimation as a CS problem [17,18] and also name the estimation methods as CCE. Sparse channel estimation methods [6–12] have been intensively studied in recent years. However, all of the proposed methods are based on the Nyquist rate sampling. The estimation methods can be classified by two types: one is mixed-norm-based convex relation algorithm, for example, least absolute shrinkage and selection operator [25] and Dantzig selector (DS) [26], and the other is greedy iterative algorithm, for example, orthogonal matching pursuit [27] and compressive sampling matching pursuit [28]. Considering the sub-Nyquist sampling, CCE method is implemented by DS algorithm, which is termed as CCE-DS. The flowchart of CCE-DS is shown in Table I. 3.4. Lower bound of compressive channel estimator To evaluate the performance of the proposed methods, we give lower bound on the basis of sub-Nyquist ADC sampling. Consider the sub-Nyquist ADC sampling, which is shown in Figure 3; the equivalent training matrix is ‚ 2 C M L , and the discrete received signal is y 2 C M 1 .
hS_ADC D
arg minK r ‚ K hK 2 ;
K
0;
= K (27) where hK contains the K dominant channel taps of h and ‚ K is the sub-matrix constructed from the K columns of X. In the following, the lower bound of channel estimator hS_ADC is obtained by average mean square error (MSE), given by o 1 n H ‚ ‚ z E kh hS_ADC k22 D E ‚ H K K K 1 H ‚H D Trace ‚ H K ‚ K K zz 1 ‚ K ‚ H K ‚ K 1 ‚ D n2 Trace ‚ H K K
2K2 o n n Trace ‚ H K ‚ K
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4. NUMERICAL SIMULATIONS In this section, we will compare the performance of the proposed estimator with least square (LS)-based linear estimator by numerical results on the basis of 1000 independent Monte Carlo runs. The broadband bandwidth is set to be W =2 D 100 MHz and channel delay spread max D 0:5 s. Hereby, the sampling length of channel vector h is set to be L D 100, and the number of dominant channel taps of h is K D 4 where the positions of Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
Compressive channel estimation with low-speed ADC
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dominant taps are generated following Gaussian distribution and subject to khk22 D 1. The Nyquist sampling rate of high-speed ADC is W D 200 MHz, and sub-Nyquist sampling rate of low-speed ADC is R D ˛W , where ˛ is a sub-Nyquist sampling factor. In this simulation, we consider the sub-Nyquist sampling factor to be ˛ D 0:4; 0:6, and 0:8. The length of training sequence is N D 120. The received signal-to-noise ratio is defined as 10 log.P =n2 /. Channel estimators hO are evaluated by average MSE, which is defined by
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where h and hO denote the channel vector and its estimator, respectively, and L is the sampling length. In Figure 6, we compare the average MSE performance of the proposed channel estimator and the LS channel estimator on the overall channel taps. It is found that the sub-Nyquist sampling rate-based compressive channel estimators are better than Nyquist rate sampling-based LS channel estimator. For the sparse multipath channel, the proposed method can estimate the dominant channel taps robustly, whereas the non-dominant channel taps are forced to zeros. Unlike the proposed method, LS-based channel estimator has to estimate all the channel taps uniformly even if the channel taps are equal to zero. It is worth mentioning that the LS channel estimator cannot work well on sub-Nyquist rate sampling, as shown in the purple curve in Figure 6; when sub-Nyquist rate R D 0:8 W is used for LS channel estimator, the average MSE performance has been significantly degraded. In Figure 7, the comparison of average MSE performance is taken on dominant channel taps. It is found that when the sub-Nyquist sampling rate R D 0:8 W, the proposed channel estimator can achieve the same average MSE performance as Nyquist sampling rate LS channel estimator. Even for the cases when R D 0:4 W and R D 0:6 W, the performances of the proposed estimator are also Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
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Proposed (DS, K=2) Proposed (DS, K=4) Proposed (DS, K=6) Proposed (DS, K=8) LS (R=0.4W) LS (R=W) Lower bound (known channel, R=0.4W)
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Figure 6. Average MSE performance of overall channel taps.
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close to LS-based one. From Figures 6 and 7, it can be concluded that the proposed channel estimator can achieve almost the same high-resolution channel estimation as the Nyquist sampling rate LS channel estimator.
Compressive channel estimation with low-speed ADC
To further confirm the advantage of the proposed method, we vary the channel sparsity, and the average MSE performances are evaluated and shown in Figures 8 and 9. The number of nonzero channel taps is set to be K D 2; 4; 6; and 8, respectively. And the sub-Nyquist sampling rate is set to be R D 0:4 W. Figure 8 depicts the average MSE performance of the overall channel taps. It is observed that sub-Nyquist sampling-based CCE method can achieve better estimation than Nyquist sampling-based LS method. In addition, as shown in the blue curves in Figure 8, the proposed method can robustly estimate the channels with different channel sparsity under the sub-Nyquist sampling rate. It is worth mentioning that the sparser the channel is, the better estimation performance can be achieved by the proposed method. However, the LS channel estimator can obtain robust performance only under the Nyquist sampling rate; when sub-Nyquist sampling rate is used, the performance is significantly degraded. Figure 9 shows the average MSE performance of the dominant channel taps. It is shown that the proposed subNyquist sampling rate-based channel estimator can achieve close performance as the Nyquist sampling rate-based LS channel estimator.
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5. CONCLUSION Broadband transmission poses a big challenge on ADC sampling capability in the next-generation wireless communication systems. To avoid the high-speed ADC sampling at receiver, we consider low-speed sampling ADC at the receiver in this paper. The channel estimation has been formulated as a CS problem, and a CCE method using DS algorithm has been proposed. By comparing with traditional Nyquist sampling rate LS channel estimation, the proposed method can reduce sampling rate while achieving similarly good estimation performance. The effectiveness of the proposed method has been verified by the simulation results.
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ACKNOWLEDGEMENTS The authors would like to thank the anonymous reviewers for their kind and constructive comments, which helped much to improve the quality of the paper. This work was supported by the Japan Society for the Promotion of Science postdoctoral fellowship.
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REFERENCES 1. Adachi F, Kudoh E. New direction of broadband wireless technology. Wireless Communications and Mobile Computing 2007; 7(8): 969–983. 2. He SB, Chen JM, Sun YX, Yau D, Yip NK. On optimal information capture by energy-constrained mobile
13.
sensors. IEEE Transactions on Vehicular Technology 2010; 59(5): 2472–2484. Chen JM, Li JK, He SB, Sun YX, Chen HH. Energyefficient coverage based on probabilistic sensing model in wireless sensor networks. IEEE Communication Letters 2010; 14(9): 833–835. Chen H, Wang G, Wang Z, So H, Poor H. Nonline-of-sight node localization based on semi-definite programming in wireless sensor networks. IEEE Transactions on Wireless Communications 2012; 11(1): 108–116. Shi Q, He C, Chen H, Jiang L. Distributed wireless sensor network localization via sequential greedy optimization algorithm. IEEE Transactions on Signal Processing 2010; 58(6): 3328–3340. Cotter SF, Rao BD. Sparse channel estimation via matching pursuit with application to equalization. IEEE Transactions on Communications 2002; 50(3): 374–377. Carbonelli C, Vedantam S, Mitra U. Sparse channel estimation with zero tap detection. IEEE Transactions on Wireless Communications 2007; 6(5): 1743–1753. Bajwa UW, Haupt J, Raz G, Nowak R. Compressed channel sensing, In 42th Annual Conference on Information Sciences and Systems (CISS), Princeton, NJ, USA, March 19–21, 2008; 5–10. Taubock G, Hlawatsch F. A compressed sensing technique for OFDM channel estimation in mobile environments: Exploiting channel sparsity for reducing pilots, In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, Nevada, USA, March 31–April 4, 2008; 1–5. Gui G, Wan Q, Huang AM, Chen ZX. Sparse multipath channel estimation using Dantzig selector algorithm, In The 12th International Symposium on Wireless Personal Multimedia Communications (WPMC), Sendai, Japan, September 7–10, 2009; 1–5. Berger CR, Zhou S, Preisig JC, Willett P. Sparse channel estimation for multicarrier underwater acoustic communication: from subspace methods to compressed sensing. IEEE Transactions on Signal Processing 2010; 58(3): 1708–1721. Gui G, Wan Q, Peng W, Adachi F. Sparse multipath channel estimation using compressive sampling matching pursuit algorithm, In 7th IEEE Vehicular Technology Society Asia Pacific Wireless Communication Symposium (APWCS), Kaohsiung, Taiwan, May 20–21, 2010; 1–5. Gui G, Peng W, Adachi F. Improved adaptive sparse channel estimation based on the least mean square algorithm, In IEEE Wireless Communications and Networking Conference (WCNC), Shanghai, China, April 7–10, 2013; 1–5.
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14. Marks RJ. Introduction to Shannon Sampling and Interpolation Theory. Spinger-Verlag: Berlin, Germany, 1991. 15. Raychaudhuri D, Mandayam NB. Frontiers of wireless and mobile communications. Proceedings of the IEEE 2012; 100(4): 824–840. 16. Candes EJ. The restricted isometry property and its implications for compressed sensing. Compte Rendus de l’Academie des Sciences 2008; Serie I(346): 589–592. 17. Candes E, Romberg J, Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transaction on Information Theory 2006; 52(2): 489–509. 18. Donoho DL. Compressed sensing. IEEE Transactions on Information Theory 2006; 52(4): 1289–1306. 19. Meng J, Yin WT, Li YY, Nguyen NT. Compressive sensing based high-resolution channel estimation for OFDM system. IEEE Journal of Selected Topics in Signal Processing 2012; 6(1): 15–25. 20. Wang Y, Yin WT. Sparse signal reconstruction via iterative support detection. SIAM Journal on Imaging Sciences 2010; 3(3): 462–491. 21. Proakis JG. Digital Communications, 4th edition. New York, NY: McGraw-Hill, 2001. 22. Tropp JA, Laska JN, Duarte MF, Romberg JK, Baraniuk RG. Beyond Nyquist: efficient sampling of sparse bandlimited signals. IEEE Transaction on Information Theory 2010; 56(1): 520–544. 23. Mishali M, Eldar YC. From theory to practice: subNyquist sampling of sparse wideband analog signals. IEEE Journal of Selected Topics in Signal Processing 2010; 4(2): 375–391. 24. Chen X, Yu Z, Hoyos S, Sadler BM, Silva-Martinez J. A sub-Nyquist rate sampling receiver exploiting compressive sensing. IEEE Transactions on Circuits and Systems I: Regular Papers 2011; 58(3): 507–520. 25. Ribshirani T. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society (B) 1996; 58(1): 267–288. 26. Candes EJ, Tao T. Rejoinder: the Dantzig selector: statistical estimation when p is much larger than n. Annals of Statistics 2007; 35(6): 2392–240. 27. Tropp JA, Gilbert AC. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transaction on Information Theory 2007; 5(12): 4655–4666. 28. Needell D, Tropp JA. CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis 2008; 26(3): 301–321.
Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
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AUTHORS’ BIOGRAPHIES Guan Gui received the Dr.Eng. degree in information and communication engineering from University of Electronic Science and Technology of China, Chengdu, China, in 2011. From March 2009 to July 2011, he was selected as outstanding doctor training candidate by the University of Electronic Science and Technology of China. From October 2009 to March 2012, with the financial support from the China scholarship council and the global center of education of Tohoku University, he joined the wireless signal processing and network laboratory (Professor Adachi’s laboratory), Department of Communication Engineering, Graduate School of Engineering, Tohoku University as a research assistant and postdoctoral research fellow, respectively. Since September 2012, he is supported by Japan society for the Promotion of Science fellowship as postdoctoral research fellow at same laboratory. His research interests are adaptive system identification, compressive sensing, sparse dictionary designing, channel estimation, and advanced wireless techniques. He is member of IEEE and IEICE. Wei Peng received her BS and MS degrees in electrical engineering from Wuhan University, Wuhan, China, in 2000 and 2003, respectively. She received the Dr.Eng. degree in electrical and electronic engineering from the University of Hong Kong, Hong Kong, in 2007. Since December 2007, she jointed Tohoku University, and worked as an assistant professor. Currently she is with the department of Huazhong University of Science and Technology, Wuhan, China, as an associate professor. Her research interests are in multiple antenna technology, cellular system, and distributed antenna network. She is an IEEE senior member. Fumiyuki Adachi received the BS and Dr.Eng. degrees in electrical engineering from Tohoku University, Sendai, Japan, in 1973 and 1984, respectively. In April 1973, he joined the Electrical Communications Laboratories of Nippon Telegraph & Telephone Corporation (now NTT) and conducted various types of research related to digital cellular mobile communications. From July 1992 to December 1999, he was with NTT Mobile Communications Network, Inc. (now NTT DoCoMo, Inc.), where he led a research group on wideband/broadband CDMA wireless access for IMT2000 and beyond. Since January 2000, he has been with
Compressive channel estimation with low-speed ADC
Tohoku University, Sendai, Japan, where he is a professor of Communication Engineering at the Graduate School of Engineering. He is currently engaged in research of gigabit wireless communication technology with a data rate above one gigabit per second, with the aim to realize the next-generation frequency and energy-efficient broadband mobile communication systems. He has been serving as the Institute of Electrical and Electronics Engineers VTS Distinguished Lecturer since 2011.From October 1984 to September 1985, he was a United Kingdom SERC Visiting Research Fellow in the Department of Electrical
G. Gui, W. Peng and F. Adachi
Engineering and Electronics at Liverpool University. He is an IEICE Fellow and was a co-recipient of the IEICE Transactions best paper of the year award 1996, 1998, and 2009 and also a recipient of Achievement Award 2003. He is an IEEE Fellow and was a co-recipient of the IEEE Vehicular Technology Transactions best paper of the year award 1980 and again 1990 and also a recipient of Avant Garde award 2000. He was a recipient of Thomson Scientific Research Front Award 2004, Ericsson Telecommunications Award 2008, and Telecom System Technology Award 2010.
Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm
